Solving Integral: $\int_{1}^{9} \frac{3x-2}{\sqrt{x}}dx = 44$

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In summary, an integral is a mathematical concept used to calculate the total value of a function within a specific range. To solve an integral, one must use integration techniques and understand the fundamental theorem of calculus. The purpose of solving an integral is to find values such as area, volume, and work done. The steps to solve an integral include expanding the expression, simplifying, integrating, substituting the limits, and calculating the difference. While a calculator can be used, it is beneficial to understand the manual steps involved.
  • #1
mothership
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[tex]
\begin{equation*}
\begin{split}
\int_{1}^{9} \frac{3x-2}{\sqrt{x}}dx &= \int_{1}^{9}\frac{3x}{x^{\frac{1}{2}}}dx - \int_{1}^{9}\frac{2}{x^{\frac{1}{2}}}dx \\ \\
&= 3\int_{1}^{9}x^{\frac{1}{2}}dx - 2\int_{1}^{9}x^{\frac{-1}{2}}dx \\ \\
&= 3 \left[ \frac{2}{3}(x^{\frac{3}{2}}) \right]_{1}^{9} - 2 \left[ 2x^{\frac{1}{2}} \right]_{1}^{9} \\ \\
&= 2 \left[9^{\frac{3}{2}} - 1 \right] - 4 \left[ 9^{\frac{1}{2}} - 1 \right] \\ \\
&= 2 \left[27 - 1 \right] - 4 \left[ 3 - 1 \right] \\ \\
&= 2*26 - 4*2 \\ \\
&= 44 \\
\end{split}
\end{equation*}
[/tex]
 
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  • #2
I don't know who Josh is and,in case you haven't figured out yourself yet,the solution you found is perfect...

Daniel.
 
  • #3
Hopefully, Josh will agree with Daniel on this issue.
 

FAQ: Solving Integral: $\int_{1}^{9} \frac{3x-2}{\sqrt{x}}dx = 44$

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a given interval. It is used to calculate the total value of a function within a specific range.

2. How do you solve an integral?

To solve an integral, you need to use integration techniques such as substitution, integration by parts, or trigonometric identities. You also need to understand the fundamental theorem of calculus, which states that integration is the inverse operation of differentiation.

3. What is the purpose of solving an integral?

The purpose of solving an integral is to find the total value of a function within a specific range. It is used in many areas of mathematics and science, such as calculating area, volume, and work done.

4. What are the steps to solve the given integral?

The steps to solve the given integral are:

  1. Expand the expression inside the integral using the distributive property.
  2. Simplify the expression by combining like terms.
  3. Use the power rule to integrate the expression.
  4. Substitute the upper and lower limits of the integral into the integrated expression.
  5. Calculate the difference between the two values to find the final answer.

5. Can this integral be solved using a calculator?

Yes, this integral can be solved using a calculator as long as it has a built-in integration function. However, it is beneficial to understand the steps involved in solving an integral manually to better understand the concept.

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